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What is the probability that among the 12 months of the year there are 3 non necessarily consecutive months containing exactly 4 birthdays?

Please help solve the following problem. Please provide step by step calculations with explanations.

Given 20 people, what is the probability that among the 12 months of the year there are 3 non necessarily consecutive months containing exactly 4 birthdays?

hints:

1. to count the number of elements of the state space, look at the following proposition:

There are ( n + r -1 choose r-1 ) distinct nonnegative integer valued vectors
(x_1, x_2, ... , x_3) satisfying

x_1 + x_2 + ... x_r = n

hint 2:

assume that each month has the same number of days, so that the probability that a birthday falls in a particular month is 1/12.

Solution Preview

Let C(n,m) denote the number to choose m from n.
1. From hint1, we put 20 people into 12 months, there are x_1 people in Jan, x_2 people in Feb, ..., x_12 people in Dec and x_1 + x_2 + ... + ...

Solution Summary

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